Superconductivity of thulium substituted clathrate hexahydrides at moderate pressure

Due to the BCS theory, hydrogen, the lightest element, would be the prospect of room-temperature superconductor after metallization, but because of the difficulty of the hydrogen metallization, the theory about hydrogen pre-compression was proposed that the hydrogen-rich compounds could be a great option for the high Tc superconductors. The superior properties of TmH6, YbH6 and LuH6 indicated the magnificent potential of heavy rare earth elements for low-pressure stability. Here, we designed XTmH12 (X = Y, Yb, Lu, and La) to obtain higher Tc while maintaining low pressure stability. Most prominently, YbTmH12 can stabilize at a pressure of 60 GPa. Compared with binary TmH6 hydride, its Tc was increased to 48 K. The results provide an effective method for the rational design of moderate pressure stabilized hydride superconductors.

Since Kamerlingh Onnes discovered that mercury (Hg) suddenly starts carrying a current without resistance at an extremely low temperature in 1911 1,2 , the achievement of room temperature superconductor is a dream for the superconductivity research.The theory that hydrogen can be metallized at high pressure was developed in 1935 and was proposed by Winger and Huntington 3 .According to the theory of superconductivity proposed by Bardeen, Cooper and Schrieffer in 1957, the transition temperature of superconductivity is proportional to the Debye temperature 4 .Due to this theory, hydrogen, the lightest element, would be the prospect of roomtemperature superconductor after metallization 5 , but because of the difficulty of the hydrogen metallization 6,7 , the theory about hydrogen pre-compression was proposed by Ashcroft that the hydrogen-rich compounds could be a great option for the high T c superconductors 8,9 .The theory of chemical pre-compression refers to the addition of other elements to the synthesized hydrogen-rich compounds at a lower pressure than synthesizing pure hydrogen 10 .Based on this conclusion, many great hydrogen-rich compounds have been designed and predicted to be potential superconductors with high T c [11][12][13] .The first successful predictions were H 3 S and LaH 10 with high T c exceeding 200 K [14][15][16] , and these predictions were successfully confirmed by experiment soon [17][18][19][20] .
Over these years, with the efforts of our researchers, almost all binary hydrides were explored, people commence the study of ternary hydride formed by adding a new element into binary hydrides.In 2019, Li 2 MgH 16 with the highest T c to date (473 K at 250 GPa), designed by filling the anti-bonding orbital of the H 2 molecular unit of MgH 16 with the element Li 21 .H-C-S compounds and Lu-N-H compounds have been widely studied for some time due to the claimed observation of room temperature superconductivity.However, there are still some controversial issues about the stoichiometry and the crystal structure [22][23][24][25] .Recently, a new kind of fluorite-type clathrate ternary hydrides AXH 8 (A = Ca, Sr, Y, La, X = B, Be, Al) in the main chain of hydrogen alloys has been predicted 26 .The most prominent, LaBeH8, is dynamically stable down to 20 GPa and has a high T c up to 185 K.The exciting thing is that the cubic clathrate superhydrides La x Y 1-x H 6,10 have been experimentally synthesized by laser heating of yttrium-lanthanum alloys, which exhibited a maximum critical temperature T c of 253 K without increasing pressure 27 .According this experiment, it is practicable to incorporate a metal element in the clathrate hydride to keep the compounds steadily.
It is a widespread attention about the prominent superconductivity of the clathrate hydrides.Clathrate hexahydrides Im-3 m-XH 6 (X = Mg, Ca, Sc, Y, La, Tm, Yb, Lu) are widespread in alkaline earth and rare earth metal superhydrides 16,[28][29][30][31][32] .In this structure, there is a body-centered cube (bcc) with center occupied by a metal atom, and there is a H 24 cage of hydrogen atoms in the void of the bcc lattice.CaH 6 and YH 6 have been experimentally synthesized with high T c s of 215 K at 172 GPa 33,34 and 227 K at 166 GPa, respectively 35 .Theoretically predicted T c s of MgH 6 , ScH 6 and LaH 6 are 260 K at 300 GPa, 147 K at 285 GPa and174 K at 100 GPa, respectively.YbH 6 and LuH 6 in full 4f.-orbitalshells are predicted to exhibit high T c superconductivity at relatively low pressures (145 K, 70 GPa vs. 273 K, 100 GPa, respectively) 32 .With unfilled 4f.orbitals, TmH 6 is stable at 50 GPa, but has a relatively low T c at 25 K.There was a report that the structures of superhydrides at low pressure could keep stable by f electrons, such as lanthanide clathrate hydrides CeH 9 36 , PrH 9 37 and NdH 9

38
. Although the filling of the metal atoms' f orbital could make the structure more stable at low pressure, according to current research results, the T c s of hydrides with unfilled 4f.orbitals are mostly very low.
In this work, we designed XTmH 12 (X = Y, Yb, Lu, and La) to obtain higher T c while maintaining low pressure stability.Most prominently, YbTmH 12 can stabilize at 60 GPa.Compared with binary TmH 6 hydride, its T c was increased to 48 K.The results provide an effective method for the rational design of moderate pressure stabilized hydride superconductors.

Results
First, we designed a series of ternary clathrate hydrides YTmH 12 based on the sodalite-like clathrate structure YLuH 12

45
. The crystal structure of Pm-3 m-YLuH 12 is shown in Fig. 1.The atoms Y, Tm, and H occupy the 1b (0.5, 0.5, 0.5), 1a (0, 0, 0), and 12 h (0.25322, 0, 0) Wyckoff positions in the crystal structure.In this structure, there are a bcc lattice of metal atoms and a H 24 cage which is formed by the hydrogen atom occupying all the tetrahedral void of the lattices, as shown in Fig. 1, and the H 24 cage is formed by six H-square and eight H-hexagon rings, with two classes of unequal H atoms.In many alkaline earth metals and rare earth metal hydrides there are this kind of structure consisting of metal atom and H 24 cage.There are two H 2 accepting electrons from the central metal atoms to form an H 4 unit, which serves as the cornerstone for the construction of a three-dimensional sodalite gabion and thus makes the structure stable.This unique structure partially occupies the degenerate orbit at the center of the region.The resulting dynamic Jahn-Teller effect contributes to enhanced electron-phonon coupling and leads to high T c superconductivity.
As is well-known, the impact of the electron correlation effects is particularly significant for 4f.systems.In our previous work, we calculated the equation of state (EOS) for YbH 2 and compared it with the experimental EOS to assess the reliability of our DFT calculations 32 .One can see that there is a good agreement between the theory and experiment for the high-pressure phase P6 3 /mmc of YbH 2 .The authors of previous work concerned with ytterbium hydrides 46 , used a Hubbard U = 5 eV for lower pressure phases and U = 0 eV for high-pressure phases to reproduce available experimental data, in clear agreement with our results.Therefore, in this work, we select GGA and U = 0 eV for calculation of 4f systems.
Next, we try to extend YTmH 12 to more compounds.In the designed XTmH 12 structure, at least one of the "pre-compressor" metal atoms is heavy rare earth element Tm, and the other element has a similar radius with Tm, including Na, K, Mg, Ca, Sr, Sc, Y, Yb, Lu, La.Then we calculated the phonon dispersion for all possible components in the pressure range of 50-200 GPa.The stability of the replaced structure is reflected in Fig. 1, we determined that only Y, Yb, Lu and La can stabilize dynamically this ternary sodalite-like clathrate structure.To determine the thermodynamic stability of these structures, we performed structure searches at a pressure of 100-200 GPa, focusing on XTmH 12 (X = Y, Yb, Lu and La) compositions with 1 to 2 formula units.As shown in Fig. 2, None of the structures Pm-3 m-XTmH 12 have the lowest enthalpy values, which means they are all metastable phases.The enthalpy of Pm-3 m-YTmH 12 and Pm-3 m-LaTmH 12 are higher than that of binary hydrides YH 6 + TmH 6 and LaH 6 + TmH 6 , respectively.Pm-3 m-YbTmH 12 and Pm-3 m-LuTmH 12 are stable compared to the binaries YbH 6 + TmH 6 and LuH 6 + TmH 6 , respectively, but their enthalpy are higher than that of the other ternary hydrides, such as C2/c-YbTmH 12 and Fd-3 m-LuTmH 12 .This means that some difficulties need to be overcome in the experiment to synthesize these structures.However, metastable stable phases can also be synthesized experimentally and even dominate over thermodynamically stable phases 47,48 .
We calculated their electronic band structures and projected density of states (PDOS).It can be clearly seen that the DOS value of the s electron of H near the Fermi surface is higher than that of the p and d electrons of the metal element.This is because H does not exist in molecular form, but forms a H 24 cage.However, compared to other high-T c hydrides with H 24 cage, such as CaH 6 , YH 6 , and LuH 6 , the DOS values of H's s electrons near the Fermi plane in these structures are not high enough, which is not good news for searching for high-temperature superconductors in hydrides.Furthermore, it is worth mentioning that the DOS of XTmH 12 has extremely high peaks near the Fermi surface.This is mainly due to the 4f orbitals from heavy rare earth elements form a set of localized and almost non-dispersive bands in XTmH 12 .These bands will appear in different positions depending on the outermost electrons of the element.The bands from Tm atom with unfilled 4f orbitals appear at the Fermi level in YTmH 12 (see Fig. 3a), and the bands from Yb atom with full-filled 4f orbitals appear about 1 eV below the Fermi level (see Fig. 3b).The bands from Tm atom with unfilled 4f orbitals also appear at the Fermi level in LaTmH 12 (see Fig. 3c), this means that the species of the other metal element has almost no effect on the energy level at which the 4f electron appears.The bands from Lu atom appear about 6 eV below the Fermi level (see Fig. 3d) because of full-filled 4f orbitals and an extra 5d electron.The f electrons can enhance the chemical compression effects from metallic elements, helping to stabilize the structure at lower pressures.
To make the prediction more reliable, evaluation of the impact of the electron correlation effects is desired.Therefore, we calculated the band structure using U = 5 eV to figure out how Hubbard-U may modify the band structure (see Fig. S1 in Supplementary Material).After considering U = 5 eV, one flat band is lifted up into the unoccupied regime.This means that the occupation of 4f states is changed, that could have substantial impact on the electron-phonon coupling physics.Future studies will focus on the impact of U to pairing strength.

Discussion
Then, we have compared the electronic density of states at the Fermi level (N Ef ) in YTmH 12 , YbTmH 12 , LuTmH 12 and LaTmH 12 and binary hexahydrides, including YH 6 , TmH 6 , YbH 6 , LuH 6 , LaH 6 , as shown in Table 1.Benefit by 4f electrons from heavy rare earth elements Tm, large electronic density of states at the Fermi level in the XTmH 12 is observed, much higher than that of the binary hexahydrides.The large H-derived electronic density  positive role in enhancing the EPC λ.However, from the aspect of partial DOS, the contribution from H to the electronic density of states at the Fermi level in XTmH 12 is not higher than the cases in binary hexahydrides.This suggests that the 4f electrons will play no role in superconductivity.The contrasting EPC λ in these clathrate hexahydrides is mainly attributed to the disparate intensity of H electrons interacting with optic phonons, rather than the contributions from global electronic structures.Papaconstantopoulos et al. apply the Gaspari-Gyorffy theory to determine that, in CaH 6 , the acoustic modes associated with Ca contribute only 7% to the total value of λ, in contrast to the optic modes associated with hydrogen which contribute 93% for the H 49 .And in LaH 10 , La has only a 2% contribution 50 .
The calculated T c s by the Allen-Dynes modified McMillan equation 51 are shown in Table 1.LaTmH 12 has a T c of 19-24 K at 170 GPa.This is not only much higher than the minimum stabilization pressure of 50 GPa for TmH 6 , but also higher than the pressure of 100 GPa for LaH 6 .LaTmH 12 requires higher pressures to remain stable, probably due to the excessive gap between the properties of La and Tm.This type of ternary clathrate structure requires the two metal elements to be close in radius and other properties to ensure H cage stability.Thus, YTmH 12 is able to stabilize at 80 GPa and exhibited T c of 40-46 K.Both the minimum stabilization pressure and T c are intermediate between the binary hydrides YH 6 and TmH 6 .Yb and Lu, which are also heavy rare earth elements adjacent to Tm, have f electrons that can similarly enhance chemical pre-compression, so the stabilization pressure of their doped structures can be reduced even further, and YbTmH 12 and LuTmH 12 can be stabilized at 60 and 70 GPa, respectively, and exhibited T c of 42-48 K. Their minimum stabilizing pressures and T c also show a pattern intermediate to that of the binary hydrides.
Charge transfer has an important effect on the structure and properties of hydrides.Table 2 shows charges transferred for all thulium substituted clathrate hexahydrides.The e represents the total remaining electrons.Negative δ mean loss of electrons, positive δ mean gain of electrons.It can be seen that La is a very good electron donor and is able to provide sufficient electrons to the surrounding H.In LaTmH 12 , each La atom can provide 2.25 electrons, ultimately making 0.21 electrons available for each H on average.However, the provision of sufficient electrons does not necessarily mean that superconductivity is favored, and may even create factors that are detrimental to superconductivity.In terms of charge transfer, the ability of Tm to provide electrons is stronger than that of Y and Yb, but unfortunately, the presence of f electrons severely constrains higher T c in thulium substituted clathrate hexahydrides.
To determine the origin of the superconductivity in these superconductors, we calculated their phonon spectrum, projected phonon density of state (PHDOS), integral EPC parameter λ and Eliashberg spectral function α 2 F(ω).The superconductivity of superconductors comes mainly from strong electron-phonon coupling (EPC).So, we can look for the frequency range in which the EPC parameter λ grows rapidly, and vibration modes in this frequency range are the key to the superconductivity of this structure.As can be easily seen in Fig. 4, λ grows rapidly in two regions: the low-frequency region and the mid-frequency region.For example, in LuTmH 12 , λ grows rapidly to 0.25 in the frequency range of 0-150 cm −1 and then grows slowly until the frequency range of 400-1000 cm −1 , where λ grows rapidly to 1.1, and then grows hardly at all (see Fig. 4a).In YbTmH 12 , λ also grows rapidly in the frequency range of 500-1000 cm −1 (see Fig. 4b).By comparing the PHDOS of different elements, we can find the reason for the rapid growth of λ.The rapid growth of λ is mainly due to the vibrations of metal atoms in the low-frequency region (red and black peaks in PHDOS), while in the mid-frequency region it is due to the vibrations of hydrogen atoms (blue peaks in PHDOS).This corresponds to the two main sources of superconductivity in such clathrate hydrides: hydrogen on the hydrogen cage and the central metallic atom.In addition to this, it can be seen in Fig. 4c that the λ of YTmH 12 grows rapidly when the frequency is 300 cm −1 .On the phonon dispersion, there are soft phonon patterns near the R direction in this frequency range.This suggests that the softening of the optical branch of the phonon spectrum is also an important source of electron-phonon coupling.In LuTmH 12 , λ grows rapidly to 0.25 in the frequency range of 0-150 cm −1 consistent with YTmH 12 (see Fig. 4d).However, in higher frequency range, λ grows much slower than that in YTmH 12 , which leads to the low T c in LaTmH 12 .
In this work, we introduce other elements to improve the superconductivity of TmH 6 , allowing the newly formed thulium substituted clathrate hexahydrides XTmH 12 (X = Y, Yb, Lu and La) to have the higher T c while
Phonon dispersion, electron-phonon coupling and Eliashberg spectral function α 2 F(ω) of XTmH 12 (X = Y, Yb, Lu and La) were calculated by the Quantum-ESPRESSO (Open-Source Package for Research in Electronic Structure, Simulation, and Optimization) 58 .With an ultra-soft potential and a cut off energy of 90 Ry, all XTmH 12 (X = Y, Yb, Lu, La) in the first Brillouin region have a k-point grid of 12 × 12 × 12 and a q-point grid of 4 × 4 × 4, respectively.The superconducting transition temperatures of XTmH

Table 2 .
Charges transferred for compounds (a) YTmH 12 , (b) YbTmH 12 , (c) LuTmH 12 , (d) LaTmH 12 .Negative δ mean loss of electrons, positive δ mean gain of electrons.maintaininglow-pressure stability.Most prominently, YbTmH 12 can be stabilized at a pressure of 60 GPa.Its T c is elevated compared to the binary TmH 6 , reaching 48 K.The results provide an effective method for the successful design of hydride superconductors at moderate pressures.